The number of small covers over cubes

Abstract

In the present paper we find a bijection between the set of small covers over an $n$–cube and the set of acyclic digraphs with $n$ labeled nodes. Using this, we give formulas of the number of small covers over an $n$–cube (generally, a product of simplices) up to Davis–Januszkiewicz equivalence classes and ${\mathbb{Z}}_2^n$–equivariant homeomorphism classes. Moreover we prove that the number of acyclic digraphs with $n$ unlabeled nodes is an upper bound of the number of small covers over an $n$–cube up to homeomorphism.